#### Abstract

In this paper we derive refinements of the Jensen type inequalities in the case of real Stieltjes measure , not necessarily positive, which are generalizations of Jensen's inequality and its reverses for positive measures. Furthermore, we investigate the exponential and logarithmic convexity of the difference between the left-hand and the right-hand side of these inequalities and give several examples of the families of functions for which the obtained results can be applied. The outcome is a new class of Cauchy-type means.

#### 1. Introduction

The authors in [1] gave some conditions on the real Stieltjes measure , not necessarily positive, under which the Jensen inequality and the converse of the Jensen inequality hold for continuous convex function . These results are derived by using the Green function defined on by The function is convex and continuous with respect to both and .

Several interesting results concerning the Jensen type inequalities have been derived by means of the function . The first one, which is stated in the following theorem, gives the conditions on the real Stieltjes measure , not necessarily positive, under which the Jensen inequality holds for continuous convex function .

Theorem 1 (see [1]). *Let be continuous function and interval such that the image of is a subset of . Let be continuous function or the function of bounded variation such that and . Then the following two statements are equivalent.*(1)*For every continuous convex function * *holds.*(2)*For all * *holds, where the function is defined in (1).**Furthermore, the statements (1) and (2) are also equivalent if we change the sign of inequality in both (2) and (3). Also note that for every continuous concave function the inequality (2) is reversed.*

*Remark 2. *Note that in the case of positive measure we get some well known results. If the function is increasing and bounded with , then inequality (2) becomes Jensen’s integral inequality. On the other hand, if the function is continuous and monotonic, and is either continuous or of bounded variation, satisfying
then inequality (2) becomes the Jensen-Steffensen inequality given by Boas in [2] (see also [3], page 59). Several other theorems concerning the inequality (2) or its reverse can be found in [3].

Similar results have also been derived for the converse of the Jensen inequality. The following theorem from [1] gives the conditions on the real Stieltjes measure , not necessarily positive such that , under which the converse of the Jensen inequality holds for continuous convex function .

Theorem 3 (see [1]). *Let be continuous function and be an interval such that the image of is a subset of . Let be such that for all . Let be continuous function or the function of bounded variation, and . Then the following two statements are equivalent.*(1)*For every continuous convex function * *holds, where .*(2)*For all * *holds, where the function is defined in (1).**Furthermore, the statements (1) and (2) are also equivalent if we change the sign of inequality in both (5) and (6).*

*Remark 4. *If we set in Theorem 3 and , the inequality (6) transforms into (see also [1])

In his papers [4, 5], Dragomir gave some inequalities concerning reverses of the Jensen inequality for positive measure. In this paper we give a generalization of those results and derive similar Jensen-type inequalities in the case of real Stieltjes measure which is not necessarily positive. Furthermore, we investigate the exponential and logarithmic convexity of the differences between the left-hand and the right-hand side of the obtained inequalities and give several examples of the families of functions for which those results can be applied.

#### 2. Main Results

Throughout this paper we will use the notation

The following result holds.

Theorem 5. *Let be a continuous function and an interval. Let be such that and for all . Let be a continuous function or a function of bounded variation such that , and let . If (6) holds, for all , then
**
holds for every continuous convex function .*

*Proof. *Let (6) hold for all . Then from Theorem 3 it follows that for every continuous convex function we have
By the convexity of (as in [4]), we have the gradient inequality
for any . If we multiply this inequality with , we get
Analogously, we get
Adding (12) to (13) and dividing by , we get that
holds for any . Substituting with , we obtain the first inequality in (9).

To prove the second inequality in (9), it is enough to notice that the function is concave on , so for every inequality is valid. That completes the proof of (9).

If we set that and , we get the following result.

Corollary 6. *Let be continuous function and (where ) an interval such that the image of is a subset of . Let be a continuous function or a function of bounded variation such that , and let . If (7) holds for all , then for every continuous convex function ,
**
provided that and are finite.*

Theorem 7. *Let be a continuous function and an interval. Let be such that for all . Let be a continuous function or a function of bounded variation such that , and let . If (6) holds for all , then for every continuous convex function we have
**
where is defined by
**
If , then we also have
*

*Proof. *Let (6) hold for all . Then from Theorem 3 it follows that
holds for every continuous convex function . If , it is easy to verify that the term on the right side of (19) is equal to
and the term in the square brackets in (20) is equal to , where the function is defined in (24). As in [5], it follows that
and since
the inequality (16) follows. For or , the inequality (16) is obvious. The inequality in (18) follows directly from the proof of inequality (16).

If we set that and , we get the following result.

Corollary 8. *Let be a continuous function and (where ) an interval such that the image of is a subset of . Let be a continuous function or a function of bounded variation such that , and let . If (7) holds for all , then
**
where is defined by
**
holds for every continuous convex function , provided that and are finite. If , then we also have
*

Theorem 9. *Let be a continuous function and an interval. Let be such that for all . Let be a continuous function or a function of bounded variation such that , and let . If (6) holds for all , then for every continuous convex function we have
*

In order to prove the theorem, we need the following lemma, which is a special case of [6, page 717, Theorem 1] for .

Lemma 10. *Let be a convex function on , and such that . Then
*

*Proof of Theorem 9. *Let (6) hold for all . Then from Theorem 3 it follows that
holds for every continuous convex function . Denote ; so we have and . As in [7], by Lemma 10 we get the following:

In order to prove the second inequality, it is easy to verify that
Since is a convex function on , by the gradient inequality we have (as in [5])
and we finally obtain
so the second inequality in (26) follows.

If we set that and , we get the following result.

Corollary 11. *Let be a continuous function and (where ) an interval such that the image of is a subset of . Let be a continuous function or a function of bounded variation such that , and let . If (7) holds for all , then for every continuous convex function we have
**
provided that and are finite.*

*Remark 12. *If in Theorem 5, Corollary 6, Theorem 7, Corollary 8, Theorem 9, and Corollary 11 we also request that (3) holds for all , then by applying Theorem 1 we will get that
also holds for every continuous convex function .

*Remark 13. *Result given in Theorem 5 together with Remark 12 represents a generalization of [4, Theorem 2], result given in Theorem 7 together with Remark 12 represents a generalization of [5, Theorem 2], and result given in Theorem 9 with Remark 12 represents a generalization of [5, Theorem 3].

#### 3. Mean-Value Theorems

As in the previous section, let be a continuous function and an interval. Let be such that for all . Let be a continuous function or a function of bounded variation such that , and let .

Motivated by the inequalities (9) and (26) for continuous convex function , we define functionals and by

If (6) holds for all , then from Theorems 5 and 9 it follows that . If the function is concave, that is, if (6) holds with reversed inequality sign for all , then we have .

Now we give two mean-value theorems for the functionals . In the following, will denote the function defined by on whatever domain we need.

Theorem 14. *Let be a continuous function and an interval. Let be such that for all . Let , . Let be a continuous function or a function of bounded variation such that and , and let and be functionals defined in (35) and (36), respectively. Then there exist such that the following equalities hold:
*

*Proof. *We only give the proof for the functional , since both proofs have the same concept. Since , there exist and . It is easy to verify that the functions defined by
are continuous and convex, and therefore , . This implies that
Hence, as the function is continuous, there exists such that
The proof for the functional is analogous.

Theorem 15. *Let be a continuous function and an interval. Let be such that for all . Let , . Let be a continuous function or a function of bounded variation such that and , and let and be functionals defined in (35) and (36), respectively. Then there exist such that the following equalities hold
**
provided that the denominators are nonzero.*

*Proof. *We give the proof for the functional . Define the function as a linear combination of functions and
Now, applying previous theorem on our function , we obtain that there exists such that
Since (otherwise we would have a contradiction with ), we get
The proof for the functional is analogous.

*Remark 16. *If the inverse of the function exists, then (41) gives

*Remark 17. *Theorems 14 and 15 also hold if we set and ; that is, if we extend the image of the function to the entire interval , provided that , , , and are finite.

#### 4. -Exponential Convexity

At the beginning of this section, let us recall some definitions and facts about exponentially convex functions (for instance, see [8] or [9]).

*Definition 18. *A function is -exponentially convex in the Jensen sense on if
holds for all and , .

A function is -exponentially convex if it is -exponentially convex in the Jensen sense and continuous on .

*Remark 19. *We can see from the definition that -exponentially convex functions in the Jensen sense are in fact nonnegative functions. Also, -exponentially convex functions in the Jensen sense are -exponentially convex in the Jensen sense for every , .

By definition of the positive semidefinite matrices and some basic linear algebra, we have the following result.

Lemma 20. *If is an -exponentially convex function in the Jensen sense, then the matrix is positive semidefinite for all , . Particularly, for all , .*

*Definition 21. *A function is exponentially convex in the Jensen sense on , if it is -exponentially convex in the Jensen sense for all .

A function is exponentially convex if it is exponentially convex in the Jensen sense and continuous.

*Remark 22. *Some examples of exponentially convex functions are as follows (see [10]):(i) defined by , where and ,(ii) defined by , where ,(iii) defined by , where .

*Remark 23. *It is known that a function is log-convex in the Jensen sense on if and only if the relation
holds for every and . It follows that a positive function is log-convex in the Jensen sense if and only if it is -exponentially convex in the Jensen sense. Also, using basic theory of convex functions, it follows that a positive function is log-convex if and only if it is -exponentially convex.

The following lemma is equivalent to the definition of convex function (see [3], page 2).

Lemma 24. *If are such that , then the function is convex if and only if the following inequality holds:
*

We will also need the following result (see [3], page 2).

Lemma 25. *If is a convex function and are such that , , , and , then the following inequality is valid:
**
If the function is concave, then the reverse inequality in (49) holds.*

When dealing with functions with different degrees of smoothness, divided differences are found to be very useful.

*Definition 26. *The second order divided difference of a function at mutually different points is defined recursively by

*Remark 27. *The value is independent of the order of the points , and . This definition may be extended to include the case in which some or all of the points coincide (see [3], page 16). Taking the limit in (50), we get
provided that exists. Furthermore, taking the limits , in (50), we get
provided that exists.

A function is convex if and only if for every choice of three mutually different points holds.

Now, we use an idea from [10] to give an elegant method of producing -exponentially convex functions and exponentially convex functions by applying functionals to a given family of functions with the same property.

For the rest of this section we assume that (6) holds, so from Theorems 5 and 9 it follows that under the appropriate assumptions on functions , , and .

Theorem 28. *Let be a continuous function and an interval. Let be such that for all . Let (where is an interval in ) be a family of functions , , such that the function is -exponentially convex in the Jensen sense on for every three mutually different points . Let be a continuous function or a function of bounded variation such that , let , and let and be linear functionals defined in (35) and (36), respectively. Then the function is -exponentially convex in the Jensen sense on . If the function is continuous on , then it is -exponentially convex on .*

*Proof. *For we define the function
where , , , and . Since is -exponentially convex in the Jensen sense by assumption, for every three mutually different points , we have
It follows that is convex (and continuous) function on , so
and hence
We conclude that the function is -exponentially convex on in the Jensen sense.

If the function is also continuous on , then it is -exponentially convex by definition.

The following corollary is an immediate consequence of the above theorem.

Corollary 29. *Let be a continuous function and an interval. Let be such that for all . Let (where is an interval in ) be a family of functions , , such that the function is exponentially convex in the Jensen sense on for every three mutually different points . Let be a continuous function or a function of bounded variation such that , let , and let and be linear functionals defined in (35) and (36), respectively. Then the function is exponentially convex in the Jensen sense on . If the function is continuous on , then it is exponentially convex on .*

Corollary 30. *Let be a continuous function and an interval. Let be such that for all . Let (where is an interval in ) be a family of functions , , such that the function is -exponentially convex in the Jensen sense on for every three mutually different points . Let be a continuous function or a function of bounded variation such that , let , and let and be linear functionals defined in (35) and (36), respectively.**Then the following statements hold.*(i)*If the function is continuous on , then it is -exponentially convex on . If is additionally strictly positive, then it is also log-convex on , and for such that , we have
*(ii)*If the function *